On maximal hyperplane sections of the unit ball of $$l_p^n$$ for $$p>2$$
Year: 2024
Type: article
Abstract: The maximal hyperplane section of the $$l_\infty ^n$$ l ∞ n -ball, i.e. of the n-cube, is the one perpendicular to $$\frac{1}{\sqrt{2}} (1,1,0 ,\ldots ,0)$$ 1 2 ( 1 , 1 , 0 , … , 0 ) , as shown by Ball. Eskenazis, Nayar and Tkocz extended this result to the $$l_p^n$$ l p n -balls for very large $$p \ge 10^{15}$$ p ≥ 10 15 . By Oleszkiewicz, Ball’s result does not transfer to $$l_p^n$$ l p n for $$2< p < p_0 \simeq 26.265$$ 2 < p < p 0 ≃ 26.265 . Then the hyperplane section perpendicular to the main diagonal yields a counterexample for large dimensions n. Suppose that $$p_0 \le p < \infty $$ p 0 ≤ p < ∞ . We show that the analogue of Ball’s result holds in $$l_p^n$$ l p n -balls for all hyperplanes with normal unit vectors a, if all coordinates of a have modulus $$\le \frac{1}{\sqrt{2}}$$ ≤ 1 2 and p has distance $$\ge 2^{-p}$$ ≥ 2 - p to the even integers. Under similar assumptions, we give a Gaussian upper bound for $$20< p < p_0$$ 20 < p < p 0 .
Source: Advances in Operator Theory
Author: Hermann König
Institution: Christian-Albrechts-Universität zu Kiel
Language: English
Topic: Advanced Banach Space Theory
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $2,990
OpenAlex ID: W4404593024